Vectors maths question

a = 3i + 4j - k
b = i - j - k

Find the vector with the magnitude of b and direction of a.

The magnitude of "b" is sqrt(3).

A unit vector in the direction of "a" is

( 3/sqrt(26), 4/sqrt(26), -1/sqrt(26) )

Multiply the scalar magnitude by the unit vector to get the resulting vector:

( sqrt(27/26), sqrt(48/26), -sqrt(3/26) )

magnitude of b is sqrt(3)
magnitude of a is sqrt(3^2+4^2 + 1) = sqrt(26)
so the scale factor for a is sqrt(3)/sqrt(26)
the vector is
a' = sqrt(3)/sqrt(26) (3i+4j-k)

V=pi+qj+rk where p,q,and r are constants.

|b|= sqrt(1+1+1)=sqrt(3)

|V|=sqrt(p^2+q^2+r^2)

VXa=0
|i j k|
|p q r|
|3 4 -1|
then:
-q-4r=0
-p-3r=0
4p-3q=0

therefore, p=-3r and q=-4r
substitute in |V|=sqrt(3) to find r
|V|=sqrt(9r^2+16r^2+r^2)=sqrt(26)r----…

p=-3r and q=-4r and r=sqrt(3/26) and V=pi+qj+rk